Northern California Symplectic Geometry
Seminar, 2021-2022
Berkeley - Davis - Santa Cruz - Stanford
The Northern California Symplectic Geometry Seminar usually meets on the first Monday of each month. Established in 1992,
the Andreas Floer Memorial Lecture usually takes place in October, during the first meeting of the seminar.
For slides or videos of the talks (when available), please click on link after the title of the talk.
Monday, Nov 1st, 2021, @Stanford (virtually)
poster
1pm-2pm (Pacific Time)
Yaniv Ganor (Technion),
Big Fiber Theorems and Ideal-Valued Measures in Symplectic Topology; slides and
video.
Abstract: In various areas of mathematics there exist "big fiber theorems", these are theorems of the following type: "For any map in a certain class, there exists a 'big' fiber", where the class of maps and the notion of size changes from case to case. We will discuss three examples of such theorems, coming from combinatorics, topology and symplectic topology from a unified viewpoint provided by Gromov's notion of ideal-valued measures. We adapt the latter notion to the realm of symplectic topology, using an enhancement of Varolgunes’ relative symplectic cohomology to include cohomology of pairs. This allows us to prove symplectic analogues for the first two theorems, yielding new symplectic rigidity results.
Necessary preliminaries will be explained.
The talk is based on a joint work with Adi Dickstein, Leonid Polterovich and Frol Zapolsky.
2:30pm-3:30pm (Pacific Time) on ZOOM
Andreas Floer Memorial Lecture Mohammed Abouzaid (Columbia University),
Complex cobordism and Hamiltonian fibrations; slides and
video.
Abstract: I will discuss joint work with McLean and Smith, lifting the
results of Seidel, Lalonde, and McDuff concerning the topology of
Hamiltonian fibrations over the 2-sphere from rational cohomology to
complex cobordism. In addition to the use of Morava K-theory (as in
the recent work with Blumberg on the Arnold Conjecture), the essential
new ingredient is the construction of global Kuranishi charts for
genus 0 pseudo-holomorphic curves; i.e. their realisation as quotients
of zero loci of sections of equivariant vector bundles on manifolds.
Monday, Dec 6, 2021, @Berkeley (virtually)
1pm-2pm (Pacific Time) on ZOOM
Anthony Bloch (University of Michigan),
Total positivity, gradient and Hamiltonian flows;
video.
Abstract:
We discuss the totally nonnegative part of an adjoint orbit of the unitary group and the notion of a twist map on such an orbit. We describe the dynamics of positivity preserving gradient flows with respect to various metrics on such an orbit and applications to the dynamics of the integrable Toda lattice flow and symmetric Toda flow, and to the topology of totally nonnegative flag varieties and amplituhedra. (joint work with Steven Karp)
2:30pm-3:30pm (Pacific Time) on ZOOM
Kyler Siegel (USC),
Symplectic capacities, unperturbed curves, and convex toric domains;
video.
Abstract: This is a progress report on the speaker's program to study higher dimensional symplectic embeddings by constructing and computing new symplectic capacities. After recalling the relevant background, I will discuss a new approach to defining certain capacities without Hamiltonian or virtual perturbations. Using this we derive formulas in terms of discrete isoperimetric problems, and use these to give sharp obstructions for various embedding problems.
Monday, Mar 7, 2022, @Stanford (virtually)
1pm-2pm (Pacific Time) on ZOOM
Aleksander Doan (Columbia),
Holomorphic Floer theory and the Fueter equationslides and
video.
Abstract: I will discuss an idea of constructing a category associated with a pair of holomorphic Lagrangians in a hyperkahler manifold, or, more generally, a manifold equipped with a triple of almost complex structures I,J,K satisfying the quaternionic relation IJ =-JI= K. This category can be seen as an infinite-dimensional version of the Fukaya-Seidel category associated with a Lefschetz fibration. While many analytic aspects of this proposal remain unexplored, I will argue that in the case of the cotangent bundle of a Lefschetz fibration, our construction recovers the Fukaya-Seidel category.
This talk is based on joint work with Semon Rezchikov, and builds on earlier ideas of Haydys, Gaiotto-Moore-Witten, and Kapranov-Kontsevich-Soibelman.
2:30pm-3:30pm (Pacific Time) on ZOOM
Oleg Lazarev (UMass Boston),
Localization and flexibilization in symplectic geometryslides and
video.
Abstract: Localization is an important construction in algebra and topology that allows one to study global phenomena a single prime at a time. Flexibilization is an operation in symplectic topology introduced by Cieliebak and Eliashberg that makes any two symplectic manifolds that are diffeomorphic (plus a bit of tangent bundle data) become symplectomorphic. In this talk, I will explain that it is fruitful to view flexibilization as a localization (at the 'prime' zero ). Building on work of Abouzaid and Seidel, l will also give examples of new localization functors of symplectic manifolds (up to stabilization and subcriticals) that interpolate between flexible and rigid symplectic geometry and can be viewed as symplectic analogs of topological localization of Sullivan, Quillen, and Bousfield.
This talk is based on joint work with Z. Sylvan and H. Tanaka.
Monday, May 2nd, 2022, @ Berkeley, in person, room 748 Evans;
2:35-2:55pm 748 Evans,
Yuan Yao (UC Berkeley)
Cascades, gluing, and ECH in the Morse-Bott setting
Abstract:
After a quick overview of Embedded Contact Homology (ECH), we will describe a strategy to compute ECH in the Morse-Bott setting. The strategy is to show curves counted by ECH (/of ECH index one) degenerate into ECH index one cascades as the contact form degenerates, and show cascades of ECH index one are in one-to-one correspondence with ECH index one curves via a gluing theorem. We give some instances of where this strategy works, and discuss some limitation of this approach to Morse-Bott ECH (mostly having to do with transversality issues).
3:00-3:20pm 748 Evans,
Luya Wang (UC Berkeley)
A connected sum formula for embedded contact homology
Abstract:
The contact connected sum is a well-understood operation for contact manifolds. I will discuss work in progress on how pseudo-holomorphic curves behave in the symplectization of the 3-dimensional contact connected sum, and as a result a connected sum formula for embedded contact homology.
3:45-4:05pm 748 Evans,
Anastasiia Sharipova (Penn State)
Viterbo’s Conjecture for Certain Hamiltonians in Classical Mechanics
Abstract:
We consider some particular cases of Viterbo’s conjecture relating volumes of convex bodies and actions of closed characteristics on their boundaries, focusing on the case of a Hamiltonian of classical mechanical type, splitting into summands depending on the coordinates and the momentum separately and establish the conjecture in several particular cases.
4:10-4:30pm 748 Evans,
Dylan Cant (Stanford)
Riemann-Roch on Surfaces with Boundary Punctures
Abstract:
Discussion of the Riemann-Roch formula computing the Fredholm index for asymptotically non-degenerate Cauchy-Riemann operators on surfaces with boundary punctures.
4:35-4:55pm 748 Evans,
James Hughes (UC Davis)
Legendrian Weaves and Lagrangian Fillings in D-type
Abstract:
Given a Legendrian link L in the contact 3-sphere, we can hope to classify the set of Lagrangian fillings of L, i.e. exact Lagrangian surfaces in the symplectic 4-ball with boundary equal to L. Recent work, including the development of the Legendrian weave calculus of Casals and Zaslow, has led to a conjectural ADE classification of Lagrangian fillings of Legendrian links. In this talk, I will introduce these Legendrian weaves and use them to present a construction of the conjectured number of fillings for D-type Legendrian links. Time permitting, I will also mention some additional applications of weaves to understanding actions of Legendrian loop isotopies on the set of Lagrangian fillings.